Determinant of n-th order and its elementary properties, expansion by a row or column,
statement of Laplace expansion, determinant of a product, statement of Cauchy-
Binet theorem, inverse through classical adjoint, Cramers rule, determinant of a partitioned
matrix, Idempotent matrices.
Norm and inner product on Rn and Cn, norm induced by an inner product, Orthonormal
basis, Gram-Schmidt orthogonalization starting from any finite set of vectors,
orthogonal complement, orthogonal projection into a subspace, orthogonal projector
into the column space of A, orthogonal and unitary matrices. Characteristic roots,
relation between characteristic polynomials of AB and BA when AB is square,
Cayley-Hamilton theorem, idea of minimal polynomial, eigenvectors, algebraic and
geometric multiplicities, characterization of diagonalizable matrices, spectral representation
of Hermitian and real symmetric matrices, singular value decomposition.
Quadratic form, category of a quadratic form, use in classification of conics, Lagranges
reduction to diagonal form, rank and signature, Sylvesters law, determinant
criteria for n.n.d. and p.d. quadratic forms, Hadamards inequality,
extrema of a p. d. quadratic form, simultaneous diagonalization of two quadratic
forms one of which is p.d., simultaneous orthogonal diagonalization of commuting
real symmetric matrices, square-root method.
Note: Geometric meaning of various concepts like subspace and flat, linear independence,
projection, determinant (as volume), inner product, norm, orthogonality,
orthogonal projection, and eigenvector should be discussed. Only finite-dimensional
vector spaces to be covered.
Reference Texts:
(a) C. R. Rao: Linear Statistical Inference and its Applications.
(b) A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
(c) K. Hoffman and R. Kunze: Linear Algebra.
(d) F. E. Hohn: Elementary Matrix Algebra.
(e) P. R. Halmos: Finite Dimensional Vector Spaces.
(f) S. Axler: Linear Algebra Done Right!
(g) H. Helson: Linear Algebra.
R Bapat: Linear Algebra and Linear Models.
(i) R. A. Horn and C. R. Johnson: Matrix Analysis.
(j) M. Artin: Algebra.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/LAlg2.html
- Teacher: B V Rajarama Bhat